Kaon Physics

Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations

Kaon physics . . . or an introduction to ﬂavor oscillations and CP violation

Harri Waltari

University of Helsinki & Helsinki Institute of Physics University of Southampton & Rutherford Appleton Laboratory

Autumn 2018

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations Contents

Kaons are the lightest strange mesons and relatively long-lived. They are from the experimental point of view the easiest platform to study many of the interesting features of flavor physics with high precision. In this lecture we shall study the flavor and CP eigenstates of kaons study how CP violation emerges in kaon physics 0 study the flavor oscillations between K 0 and K This lecture corresponds to chapters 14.4, 14.5.1, 14.5.2 and 14.5.4 of Thomson’s book.

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations 0 Weak interactions mix the neutral kaon states K 0 and K

Neutral kaons are produced in two diﬀerent quark compositions K 0 0 (ds) and K (ds) — notice that for historical reasons, the strangeness quantum number is so deﬁned that S(K 0) = +1, i.e. the antiquark has positive strangeness As neutral kaons propagate, they can transform between each other 0 through box diagrams, so neither K 0 nor K is an eigenstate of the full Hamiltonian

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations The physical eigenstates have diﬀering lifetimes

Experimentally we observe that there are two different neutral kaon states with (almost) the same mass (close to 500 MeV) but different lifetimes, one having τ ' 0.1 ns and the other with τ ' 50 ns, these are called KS and KL (for short and long) We know that in the leptonic sector CP is (at least within current experimental precision) conserved so we may expect the CP eigenstates to be (at least close to) the stationary states 0 0 Kaons are JP = 0− mesons so P|K 0i = −|K 0i and P|K i = −|K i The flavor eigenstates have opposite flavor contents so 0 0 C|K 0i = eiζ |K i and C|K i = e−iζ |K 0i, where ζ is an unobservable phase If we choose ζ = π (which is a common convention) 0 0 C|K 0i = −|K i and C|K i = −|K 0i 0 0 Hence CP|K 0i = |K i and CP|K i = |K 0i

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations 0 The CP eigenstates are combinations of K 0 and K

0 From the CP properties of K 0 and K it is easy to construct the CP eigenstates: CP eigenstates of neutral kaons 0 |K i = √1 (|K 0i + |K i) with CP = +1 and 1 2 0 |K i = √1 (|K 0i − |K i) with CP = −1 2 2

In weak interactions CP is nearly conserved so these almost coincide with the physical eigenstates KS and KL

On the other hand flavor eigenstates are linear combinations of |K1i and |K2i but this combination will have a non-trivial time evolution As we shall see, some of the decays are spesific to flavor eigenstates and some to CP eigenstates, which will give us interesting probes on the kaon system

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations

Decays to pions determine the CP properties of KS and KL

0 0 + − Experimentally we see that KS decays mainly to π π or π π , 0 0 0 + 0 − while KL decays to π π π or π π π These decay modes determine the lifetimes — for the three pion decay there is less phase space available so the decay is slower As both kaons and pions are JP = 0− mesons, the pions must be in an ` = 0 state in the decay K 0 → π0π0, π+π− to conserve angular momentum Hence the parity of the ﬁnal state is P(π0π0) = (−1)`P(π0)P(π0) = 1 · (−1) · (−1) = 1, similarly for π+π− The neutral pion is a state |π0i = √1 (uu − dd) so its C-parity is +1 2 as C transforms the pion to itself For π+π− C is equivalent to the exchange of particles and for bosons this does not change the sign 0 0 + − Hence CP(π π ) = +1 and CP(π π ) = +1 implying that KS could be identiﬁed with K1

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations

Decays to pions determine the CP properties of KS and KL

The angular momentum argument is trickier for the three pion ﬁnal state

~ We ﬁrst take the angular momentum of two pions labeling it L1 and then the angular momentum of the third with respect to the ~ center-of-mass of the pair (L2) ~ ~ ~ The total angular momentum is L = L1 + L2, which must be zero due to angular momentum conservation (the spins are all zero), giving |L1| = |L2| Hence the parity is (−1)L1 · (−1)L2 · (P(π0))3 = −1

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations

Decays to pions determine the CP properties of KS and KL

The C-parities of the three-pion ﬁnal states are +1 by the same arguments as for the two pion case Hence the CP for the three pion state is negative, which implies CP(KL) = −1

Thus |KLi can be identiﬁed with |K2i In 1964 Christenson, Cronin, Fitch and Turlay observed the decay + − KL → π π , which was the ﬁrst (and together with other similar kaon decays, the only for more than 30 years) indication of CP violation in weak interactions (Fitch and Cronin got the 1980 Nobel Prize for this discovery)

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations Particle decays can be described by complex energy eigenvalues

We shall ﬁrst look at kaon state vectors in the limit of CP conservation Kaons are produced as ﬂavor eigenstates, e.g. K 0 = √1 (|K i + |K i) 2 S L The time evolution can be properly described if we attach an imaginary part to the eigenvalue of the Hamiltonian, e.g. −im t−Γ t/2 |KS (t)i = |KS (0)ie S S −Γ t In such a case hKS (t)|KS (t)i = e S so the wave function dies out in the timescale τS = 1/ΓS

As there is a factor of 500 between the lifetimes of KS and KL, after a reasonably long propagation time a beam initially in a ﬂavor eigenstate has become a pure |KLi state

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations CP is nearly conserved — only a tiny fraction of decays are CP violating

KS decays BR KL decays BR + − + − KS → π π 69.2% KL → π π 0.20% 0 0 0 0 KS → π π 30.7% KL → π π 0.09% 0 0 0 −8 0 0 0 KS → π π π < 2.6 × 10 KL → π π π 19.5% + − 0 −7 + − 0 KS → π π π 3.5 × 10 KL → π π π 12.5% − + − + KS → π e νe 0.03% KL → π e νe 20.3% + − + − KS → π e νe 0.03% KL → π e νe 20.3%

Here the leptonic modes have equal decay rates but due to the larger total rate of KS , the branching ratios are smaller There is a clear diﬀerence in the order of magnitude of the CP violating decays, 2 orders of magnitude when kinematics are favorable and 6 orders of magnitude when they are not

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations CP is violated both in kaon mixing and decays

There are two possible sources for CP violation: The Hamiltonian can introduce CP violation in the kaon mixing so that the physical eigenstates do not coincide with the CP ones, it could also be possible for the kaon decays to be directly aﬀected by the CP violating phase In the ﬁrst case the eigenstates are parametrized as 1 1 |KS i = (|K1i + |K2i), |KLi = (|K2i + |K1i), p1 + ||2 p1 + ||2

where is a small complex parameter 0 The second case is parametrized by = Γ(K2 → ππ)/Γ(K2 → πππ) Both are nonzero and small and <(0/) ' 1.65 × 10−3

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations The Hamiltonian gets contributions from interaction potentials and decays

We shall next go through the kaon mixing and see how most of the CP violation is generated. The mixing is a second order eﬀect in weak 2 interactions, i.e. proportional to GF . The Hamiltonian equation of motion for, say, |K 0i in its rest frame ∂ 0 i 0 is i ∂t |K (t)i = (m − 2 Γ)|K (t)i The mass term is the sum of the quark masses and the potential energy of the interactions between quarks:

0 0 0 0 X hK |HW |jihj|HW |K i m = md +ms +hK |HQCD +HQED +HW |K i+ Ej − mK j

The decay rate is determined by the Fermi golden rule: P 0 2 Γ = 2π f |hf |HW |K i| ρf , where the sum goes over all possible ﬁnal states |f i and ρf is the corresponding density of states

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations Weak interactions introduce mixing terms to the Hamiltonian

The mass terms M11 and M22 are similar to the ones before, but M12 and M21 arise from the box diagrams:

0 0 ∗ X hK |HW |jihj|HW |K i M12 = M21 = Ej − mK j

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations Weak interactions introduce mixing terms to the Hamiltonian

The decay term can be written as

X 2 X 0 0 2 |hf |HW |K(t)i| = |hf |HW |(a(t)|K i + b(t)|K i)| = f f X 2 0 0 2 0 0 |a(t)| hK |HW |f ihf |HW |K i + |b(t)| hK |HW |f ihf |HW |K i+ f ∗ 0 0 ∗ 0 0 a (t)b(t)hK |HW |f ihf |HW |K i + a(t)b (t)hK |HW |f ihf |HW |K i

The ﬁrst two terms can be identiﬁed as Γ11 and Γ22, whereas the two ∗ latter ones are Γ12 and Γ21 = Γ12.

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations Diagonalizing the Hamiltonian leads to the physical eigenstates

From the deﬁnition we see that M11, M22,Γ11 and Γ22 are real, whereas M12 and Γ12 are complex (and are the source of CP violation)

CPT invariance requires M11 = M22 = M and Γ11 = Γ22 = Γ so the equation of motion simpliﬁes to ! ! ∂ a(t)|K 0i i i a(t)|K 0i M − 2 Γ M12 − 2 Γ12 i 0 = ∗ i ∗ i 0 ∂t b(t)|K i M12 − 2 Γ12 M − 2 Γ b(t)|K i We may solve the eigenvalues of the Hamiltonian giving s i i i E = M − Γ ± M∗ − Γ∗ M − Γ , ± 2 12 2 12 12 2 12 the real parts giving the masses and the imaginary parts giving the decay widths

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations

The eigenstates diﬀer slightly from |K1,2i

The eigenvectors of the Hamiltonian (i.e. the stationary states) can 0 be solved and they are √ 1 (|K 0i + ξ|K i and 1+|ξ|2 0 √ 1 (|K 0i − ξ|K i, where 1+|ξ|2

1/2 ∗ i ∗ ! M12 − 2 Γ12 ξ = i M12 − 2 Γ12

If M12 and Γ12 were real, ξ = 1 and the eigenstates would coincide with |K1,2i, however they are complex, which leads to CP violation 1− Rewriting ξ = 1+ leads to eigenstates 1 |KS i = (|K1i + |K2i) p1 + ||2 1 |KLi = (|K2i + |K1i) p1 + ||2

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations The kaon eigenstates have a tiny mass diﬀerence

From the eigenvalues we get q ∗ i ∗ i E+ − E− = 2 M12 − 2 Γ12 M12 − 2 Γ12 i We may write E± = M ± ∆M/2 − 2 (Γ ± ∆Γ/2), where ∆M = |<(E+ − E−)| and ∆Γ = ±|2=(E+ − E−)|, where the sign of ∆Γ depends on the experimental data on the lifetimes Experimentally the heavier state has the longer lifetime, so it is associated with |KLi

The states KS and KL have a tiny mass diﬀerence −15 ∆mK = 3.5 × 10 GeV (we will see soon, how can this be measured — the masses are not known to this precision)

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations 0 The kaons oscillate between K 0 and K and relax to a state with equal weights

Kaons are produced in flavor eigenstates, but since they are not the eigenstates of the Hamiltonian, they have a nontrivial time evolution, which leads to flavor oscillations The oscillations are due to the box diagrams with two W -bosons Since the KS component decays faster, the oscillations will 0 0 eventually end as the KL has fixed proportions of K and K Measuring flavor oscillations is possible because the semileptonic 0 − + decays show the flavor of the kaon: Only K → π e νe and 0 + − K → π e νe are possible, so the charge of the electron (or pion) tells us, which flavor the kaon had

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations The time evolution is dictated by the CP eigenstates

We’ll ﬁrst neglect CP violation as it is a small eﬀect. Look at a state produced as K 0: The initial state is |K 0i = √1 (|K i + |K i) 2 S L This evolves in time as 0 |K(t)i = √1 (θ |K i + θ |K i) = 1 (θ + θ )|K 0i + 1 (θ − θ )|K i 2 S S L L 2 S L 2 S L −im t−Γ t/2 Here θS,L = e S,L S,L are the phase factors for decaying eigenstates of the Hamiltonian The probability of observing a kaon as K 0 is 0 0 0 2 1 2 P(Kt=0 → Kt ) = |hK |K(t)i| = 4 |θS + θL| 0 Similarly the probability of observing a K is 0 0 0 2 1 2 P(Kt=0 → K t ) = |hK |K(t)i| = 4 |θS − θL|

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations The oscillation probability is sensitive to the mass diﬀerence

2 2 2 ∗ Expanding these gives |θS ± θL| = |θS | + |θL| ±2<(θS θL) |{z} |{z} =e−ΓS t =e−ΓLt −(Γ +Γ )t/2−i∆m t −(Γ +Γ )t/2 The last term is 2<(e S L K ) = 2e S L cos(∆mK t) Hence the oscillation period is sensitive to the kaon mass difference (like neutrino oscillations), although the amplitude is decaying rapidly The mass difference is so small that the oscillation period is longer than the KS lifetime so only the first oscillation is somewhat visible

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations The CPLEAR experiment measured kaon ﬂavor oscillations

0 Kaons were produced by pp → π+K −K 0, π−K +K , the kaon charge gives the neutral kaon ﬂavor

Figure: The CPLEAR collaboration, Phys. Rept. 403 (2004) 303

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations Cerenkov counters were used to identify kaons

The kinematics were such that pions, electrons and muons gave a Cerenkov signal, while kaons did not — helps in triggering and particle identification The low energy meant that the neutral kaons decayed within the detector, their flight distance indicating the lifetime The events of interest were the semileptonic decays of kaons, where the charge of the lepton gave the flavor of the kaon

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations Many uncertainties cancel when the relative diﬀerence in rates is considered

The probability of K 0 decaying as a K 0 is measured from events with a K − and e+ in the ﬁnal state, this will be proportional to −Γ t −Γ t −(Γ +Γ )t/2 e S + e L + 2e S L cos(∆mK t) 0 The probability of K 0 oscillating to a K before decaying is measured from events with a K − and e− in the ﬁnal state, which is −Γ t −Γ t −(Γ +Γ )t/2 proportional to e S + e L − 2e S L cos(∆mK t) When considering the ratio

0 P(K 0 → K 0) − P(K 0 → K ) A(t) = t=0 t t=0 t 0 0 0 0 P(Kt=0 → Kt ) + P(Kt=0 → K t ) uncertainties related to the production rate cancel, it is possible to 0 include also the similar rates for K

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations

The asymmetry in the decays allows us to measure ∆mK

Plugging in the expressions for the various rates leads to the expression 2e−(ΓS +ΓL)t/2 cos(∆m t) A(t) = K e−ΓS t + e−ΓLt for the asymmetry in the kaon decays The measured asymmetry is consistent with −15 ∆mK = 3.5 × 10 GeV, whereas the masses themselves have been measured with a precision of 10−5 GeV

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations

CP violation can be seen in semileptonic decays of KL

We may observe CP violation in the semileptonic decays if we look at events far from the interaction point, where the state is a pure |KLi state 1 0 0 Since |KLi = √ [(1 + )|K i − (1 − )|K i], there will be a 2(1+||2) slight diﬀerence in the rates of the semileptonic decays ± ∓ 2 We have Γ(KL → π e ν) ∝ |1 ∓ | ' 1 ∓ 2<() This allows the measurement of CP violation in terms of the asymmetry parameter (writing = ||eiφ)

− + + − Γ(KL → π e νe ) − Γ(KL → π e νe ) δ = − + + − ' 2<() = 2|| cos φ Γ(KL → π e νe ) + Γ(KL → π e νe )

Experimentally we have δ = 0.327 ± 0.012% showing again that CP −3 is violated in weak interactions (|| & 1.6 × 10 ) Next time: How to measure the full and not just its real part

H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and ﬂavor oscillations Summary

Kaons are usually produced as flavor eigenstates, they propagate nearly as CP eigenstates (modulo small CP violation) and they may decay as either of them Most of the CP violation in the neutral kaon system comes from the 0 K 0–K mixing through the box diagrams The flavor oscillations can be seen in semileptonic decays and they are sensitive to the tiny mass difference between KS and KL Also CP violation can be seen in the semileptonic decays of kaons

H. Waltari Kaon physics